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A141369 E.g.f. satisfies A(x) = exp(x*A(-x)). 10

%I #18 Feb 28 2023 06:48:21

%S 1,1,-1,-8,21,336,-1445,-35328,212009,7010560,-54073449,-2258780160,

%T 21303275389,1076400869376,-12005345614093,-712084337721344,

%U 9169911825026385,624667885401341952,-9122376282532978769,-701910552416102645760,11462725659070874233061

%N E.g.f. satisfies A(x) = exp(x*A(-x)).

%H Seiichi Manyama, <a href="/A141369/b141369.txt">Table of n, a(n) for n = 0..405</a>

%F E.g.f.: A(x) = exp(x*exp(-x*exp(x*exp(-x*exp(x*...))))).

%F a(n+1) = Sum_{i=0..n} (i+1)*(-1)^i*binomial(n,i)*a(i)*a(n-i) - from a formula given in A096538 by _Vladeta Jovovic_.

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (n-k+1)^(k-1) * k^(n-k). - _Paul D. Hanna_, Jun 13 2009

%F |a(n)| ~ c * n! / (n^(3/2) * r^n), where r = 0.5098636055230131449434409623392631606695606770070519241... is the root of the equation r*exp(1/LambertW(-I/r))/I = LambertW(-I/r), and c = 0.63217617290426743984700577681768332... if n is even, and c = 1.4315233793609300008688492299361204... if n is odd. - _Vaclav Kotesovec_, Feb 26 2014

%e E.g.f.: A(x) = 1 + x - x^2/2! - 8*x^3/3! + 21*x^4/4! + 336*x^5/5! --++ ...

%e Log(A(x)) = x - x^2 - x^3/2! + 8*x^4/3! + 21*x^4/4! - 336*x^5/5! -++- ...

%t Flatten[{1,Table[Sum[(-1)^(n-k) * Binomial[n,k] * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 26 2014 *)

%o (PARI) {a(n)=local(A=1); for(i=0, n, A=exp((-1)^(n-i)*x*A+x*O(x^n))); n!*polcoeff(A, n)}

%o (PARI) {a(n)=sum(k=0,n,(-1)^(n-k)*binomial(n,k)*(n-k+1)^(k-1)*k^(n-k))} \\ _Paul D. Hanna_, Jun 13 2009

%Y Cf. A096538.

%K sign

%O 0,4

%A _Paul D. Hanna_, Jun 28 2008

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